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Register a new userAsymptotic linking of volume-preserving actions of ${mathbb R}^k$
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We extend V. Arnolds theory of asymptotic linking for two volume preserving flows on a domain in ${mathbb R}^3$ and $S^3$ to volume preserving actions of ${mathbb R}^k$ and ${mathbb R}^ell$ on certain domains in ${mathbb R}^n$ and also to linking of a volume preserving action of ${mathbb R}^k$ with a closed oriented singular $ell$-dimensional submanifold in ${mathbb R}^n$, where $n=k+ell+1$. We also extend the Biot-Savart formula to higher dimensions.
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For the moduli space of unmarked convex $mathbb{RP}^2$ structures on the surface $S_{g,m}$ with negative Euler characteristic, we investigate the subsets of the moduli space defined by the notions like boundedness of projective invariants, area, Gromov hyperbolicity constant, quasisymmetricity constant etc. These subsets are comparable to each other. We show that the Goldman symplectic volume of the subset with certain projective invariants bounded above by $t$ and fixed boundary simple root lengths $mathbf{L}$ is bounded above by a positive polynomial of $(t,mathbf{L})$ and thus the volume of all the other subsets are finite. We show that the analog of Mumfords compactness theorem holds for the area bounded subset.
We consider the flow of closed convex hypersurfaces in Euclidean space $mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.
The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space $mathbb{R}^{3,1}$, up to orbit equivalence. Representations, up to conjugacy, of the acting groups in $O(3,1)ltimes mathbb{R}^{3,1}$ are given in both cases, proper and non-proper actions. When the action is proper, the orbits and the orbit spaces are determined.
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms and A-stable maps f and for K-equivalence with volume-preserving source diffeomorphisms and K-simple maps f. On the other hand, there are A-stable maps f with infinite-dimensional moduli space for A-equivalence with volume-preserving source diffeomorphisms.
We estimate the upper bound for the $ell^{infty}$-norm of the volume form on $mathbb{H}^2timesmathbb{H}^2timesmathbb{H}^2$ seen as a class in $H_{c}^{6}(mathrm{PSL}_{2}mathbb{R}timesmathrm{PSL}_{2}mathbb{R}timesmathrm{PSL}_{2}mathbb{R};mathbb{R})$. This gives the lower bound for the simplicial volume of closed Riemennian manifolds covered by $mathbb{H}^{2}timesmathbb{H}^{2}timesmathbb{H}^{2}$. The proof of these facts yields an algorithm to compute the lower bound of closed Riemannian manifolds covered by $big(mathbb{H}^2big)^n$.
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